The discrete Heisenberg group has growth degree four

Write $H={\mathbb{Z}}^3$ with multiplication $(a,b,c)(a^{\mathrm{\prime }},b^{\mathrm{\prime }},c^{\mathrm{\prime }})=(a+a^{\mathrm{\prime }},b+b^{\mathrm{\prime }},c+c^{\mathrm{\prime }}+a{b}^{\mathrm{\prime }})$. A direct commutator calculation gives $[(a,b,c),(a^{\mathrm{\prime }},b^{\mathrm{\prime }},c^{\mathrm{\prime }})]=(0,0,a{b}^{\mathrm{\prime }}-a^{\mathrm{\prime }}b)$, so ${\gamma }_2(H)=[H,H]=\{(0,0,c):c\in \mathbb{Z}\}$ and ${\gamma }_3(H)=1$.

The quotient ${\gamma }_1(H)\mathrm{/}{\gamma }_2(H)$ is generated by the images of $(1,0,0)$ and $(0,1,0)$ and is isomorphic to ${\mathbb{Z}}^2$, while ${\gamma }_2(H)\mathrm{/}{\gamma }_3(H)\cong \mathbb{Z}$. Therefore [L1] gives $D(H)=1\cdot 2+2\cdot 1=4$.

Put $x=(1,0,0)$, $y=(0,1,0)$ and $z=[x,y]=(0,0,1)$, using $[x,y]=xy{x}^{-1}{y}^{-1}$. Every element has the unique form $x^a{y}^b{z}^d=(a,b,ab+d)$, so $x,y$ generate $H$. A word of length at most $m$ in $x^{\pm 1},y^{\pm 1}$ has $\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid }\le m$ and $\mathrm{\mid }c\mathrm{\mid }\le m^2$: multiplication by $x^{\pm 1}$ changes only $a$, and multiplication by $y^{\pm 1}$ changes $b$ by $\pm 1$ and $c$ by $\pm a$. Thus its ball has at most $(2m+1{)}^2(2m^2+1)$ elements.

For an integer $d>0$, put $q=\lfloor \sqrt{d}\rfloor \ge 1$ and write $d=q\mathrm{\ell }+r$ with $0\le r<q$. Then $\mathrm{\ell }\le q+2$ and $z^d=[x^q,y^{\mathrm{\ell }}][x^r,y]$, a word of length at most $2q+2\mathrm{\ell }+2r+2\le 6q+6$. For $d<0$ invert a word for $z^{-d}$, and for $d=0$ use the empty word. Hence, for every integer $k\ge 1$, all the distinct elements $x^a{y}^b{z}^d$ with $\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid }\le k$ and $\mathrm{\mid }d\mathrm{\mid }\le k^2$ lie in the ball of radius $8k+6$. That ball has at least $(2k+1{)}^2(2k^2+1)$ elements.

The upper bound in step 2.2 is $O(m^4)$ for $m\ge 1$. For $m\ge 28$, take $k=\lfloor (m-6)\mathrm{/}8\rfloor \ge m\mathrm{/}16$ in step 3.1 to obtain a lower bound by a positive constant times $m^4$. These bounds prove equivalence with $m^4$ for $\{x,y\}$, and [L2] transfers that growth type to every finite generating set. Together with step 2.1 this proves the example.